We evaluate the information-theoretic achievable rates of Quantize-Map-and-Forward (QMF) relaying schemes over Gaussian $N$-relay diamond networks. Focusing on vector Gaussian quantization at the relays, our goal is to understand how close to the cutset upper bound these schemes can achieve in the context of diamond networks, and how much benefit is obtained by optimizing the quantizer distortions at the relays. First, with noise-level quantization, we point out that the worst-case gap from the cutset upper bound is $(N+\log_2 N)$ bits/s/Hz. A better universal quantization level found without using channel state information (CSI) leads to a sharpened gap of $\log_2 N + \log_2(1+N) + N\log_2(1 + 1/N)$ bits/s/Hz. On the other hand, it turns out that finding the optimal distortion levels depending on the channel gains is a non-trivial problem in the general $N$-relay setup. We manage to solve the two-relay problem and the symmetric $N$-relay problem analytically, and show the improvement via numerical evaluations both in static as well as slow-fading channels.